cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A360279 Decimal expansion of a constant related to the asymptotics of A302702.

Original entry on oeis.org

2, 1, 2, 4, 6, 0, 6, 5, 8, 3, 6, 2, 4, 2, 8, 9, 7, 9, 1, 8, 2, 7, 8, 8, 2, 5, 7, 4, 6, 9, 8, 9, 2, 4, 1, 7, 1, 6, 8, 6, 2, 5, 9, 6, 6, 4, 0, 5, 1, 0, 9, 0, 7, 2, 3, 1, 1, 2, 0, 8, 2, 0, 1, 8, 3, 1, 6, 9, 2, 8, 8, 6
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 01 2023

Keywords

Examples

			2.12460658362428979182788257469892417168625966405109072311208201831692886...

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360235, A360236, A360237.

Formula

Equals lim_{n->infinity} (A360231(n) / n!)^(1/n).
Equals lim_{n->infinity} (A302702(n) / n!)^(1/n).
Equals lim_{n->infinity} (A302703(n) / n!)^(1/n).
Equals lim_{n->infinity} (A360234(n) / n!)^(1/n).
Equals lim_{n->infinity} (A360235(n) / n!)^(1/n).
Equals lim_{n->infinity} (A360236(n) / n!)^(1/n).
Equals lim_{n->infinity} (A360237(n) / n!)^(1/n).

A302703 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 3, 21, 235, 3470, 61933, 1274893, 29423331, 747440115, 20636072811, 613611700946, 19517927805840, 660667692682175, 23699856058131981, 897955765812058192, 35832679277251514074, 1502303284645831488072, 66031982339561373164915, 3036884343153028302140119, 145885192794643951791449387
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 235*x^4 + 3470*x^5 + 61933*x^6 + 1274893*x^7 + 29423331*x^8 + 747440115*x^9 + 20636072811*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 4*x^2 + 31*x^3 + 356*x^4 + 5291*x^5 + 94592*x^6 + 1948763*x^7 + 45025516*x^8 + 1145651239*x^9 + 31696223593*x^10 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1, 2/2, 12/3, 124/4, 1780/5, 31746/6, 662144/7, 15590104/8, ...]
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  3,  21,  235,  3470,  61933,  1274893, ...];
n=1: [1, 2,  7,  48,  521,  7536, 132657,  2704342, ...];
n=2: [1, 3, 12,  82,  867, 12288, 213282,  4304877, ...];
n=3: [1, 4, 18, 124, 1283, 17828, 305056,  6094832, ...];
n=4: [1, 5, 25, 175, 1780, 24271, 409380,  8094540, ...];
n=5: [1, 6, 33, 236, 2370, 31746, 527824, 10326546, ...];
n=6: [1, 7, 42, 308, 3066, 40397, 662144, 12815839, ...];
n=7: [1, 8, 52, 392, 3882, 50384, 814300, 15590104, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+1))^(n+1):
n=0: [1, 1,  1,   3,   21,   235,    3470,    61933, ...];
n=1: [1, 2,  5,  18,  114,  1166,   16355,   283142, ...];
n=2: [1, 3, 12,  55,  354,  3372,   44463,   739917, ...];
n=3: [1, 4, 22, 124,  857,  7908,   98244,  1558788, ...];
n=4: [1, 5, 35, 235, 1780, 16501,  195980,  2955095, ...];
n=5: [1, 6, 51, 398, 3321, 31746,  368032,  5294250, ...];
n=6: [1, 7, 70, 623, 5719, 57302,  662144,  9182013, ...];
n=7: [1, 8, 92, 920, 9254, 98088, 1149804, 15590104, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A360234, A360235, A360236, A360237.
Cf. A360345, A360337.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.1246065836242897918278825..., alfa = 2.2013296851505132606640400434738193121994558898350865326..., c = 0.026186121837027622395555466054900245177877028741031867... - Vaclav Kotesovec, Oct 06 2020, updated Feb 05 2023

A360231 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n-1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 1, 6, 53, 628, 9167, 156309, 3021720, 64960004, 1532234825, 39270176511, 1085601040372, 32185085432757, 1018593646880447, 34279111177431666, 1222648239226278333, 46084480032637208699, 1830881732391546532475, 76488074741796221197580, 3352854778050665597014436
Offset: 0

Views

Author

Paul D. Hanna, Feb 02 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 53*x^4 + 628*x^5 + 9167*x^6 + 156309*x^7 + 3021720*x^8 + 64960004*x^9 + 1532234825*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 2*x^2 + 10*x^3 + 86*x^4 + 1004*x^5 + 14507*x^6 + 246218*x^7 + 4753205*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n-1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 6/3, 40/4, 430/5, 6024/6, 101549/7, 1969744/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  1,   6,   53,  628,   9167,  156309, ...];
n=1: [1, 2,  3,  14,  119, 1374,  19732,  332844, ...];
n=2: [1, 3,  6,  25,  201, 2259,  31891,  531933, ...];
n=3: [1, 4, 10,  40,  303, 3308,  45870,  756192, ...];
n=4: [1, 5, 15,  60,  430, 4551,  61930, 1008565, ...];
n=5: [1, 6, 21,  86,  588, 6024,  80373, 1292370, ...];
n=6: [1, 7, 28, 119,  784, 7770, 101549, 1611352, ...];
n=7: [1, 8, 36, 160, 1026, 9840, 125864, 1969744, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n-1))^(n+1):
n=0: [1, 1, -1,   0,   -5,   -42,   -528,   -7939, ...];
n=1: [1, 2,  1,   0,    0,     0,      0,       0, ...];
n=2: [1, 3,  6,  10,   30,   207,   2266,   31824, ...];
n=3: [1, 4, 14,  40,  141,   808,   7694,  101288, ...];
n=4: [1, 5, 25, 100,  430,  2376,  19680,  235165, ...];
n=5: [1, 6, 39, 200, 1035,  6024,  45879,  490524, ...];
n=6: [1, 7, 56, 350, 2135, 13601, 101549,  988338, ...];
n=7: [1, 8, 76, 560, 3950, 27888, 213952, 1969744, ...]; ...
to see that the main diagonals of the tables are the same:
[1, 2, 6, 40, 430, 6024, 101549, 1969744, ...].

Links

Crossrefs

Cf. A302702, A302703, A360234, A360235, A360236, A360237.
Cf. A360337, A360345.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m-1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n-1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n-1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279, alfa = 0.311338934287018467072138011497837... and c = 0.1932932528309324180094... - Vaclav Kotesovec, Feb 03 2023

A360234 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 4, 33, 414, 6750, 131963, 2957899, 73968136, 2027178710, 60143834893, 1914750144642, 64984397381766, 2339387034919340, 88976089246855623, 3563952072597604091, 149941204887915187568, 6610797722288579969347, 304837386103152855175255, 14675559490665539299350303
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 33*x^3 + 414*x^4 + 6750*x^5 + 131963*x^6 + 2957899*x^7 + 73968136*x^8 + 2027178710*x^9 + 60143834893*x^10 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 5*x^2 + 46*x^3 + 603*x^4 + 10011*x^5 + 197357*x^6 + 4444483*x^7 + 111520277*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 15/3, 184/4, 3015/5, 60066/6, 1381499/7, 35555864/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  4,  33,  414,  6750,  131963,  2957899, ...];
n=1: [1, 2,  9,  74,  910, 14592,  281827,  6261048, ...];
n=2: [1, 3, 15, 124, 1500, 23673,  451690,  9944484, ...];
n=3: [1, 4, 22, 184, 2197, 34156,  643878, 14046740, ...];
n=4: [1, 5, 30, 255, 3015, 46221,  860965, 18610170, ...];
n=5: [1, 6, 39, 338, 3969, 60066, 1105794, 23681298, ...];
n=6: [1, 7, 49, 434, 5075, 75908, 1381499, 29311192, ...];
n=7: [1, 8, 60, 544, 6350, 93984, 1691528, 35555864, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+2))^(n+1):
n=0: [1, 1,   2,    9,    74,    910,   14592,   281827, ...];
n=1: [1, 2,   7,   36,   287,   3338,   51315,   963446, ...];
n=2: [1, 3,  15,   91,   744,   8337,  122662,  2227101, ...];
n=3: [1, 4,  26,  184,  1591,  17600,  249194,  4361112, ...];
n=4: [1, 5,  40,  325,  3015,  33656,  463710,  7824385, ...];
n=5: [1, 6,  57,  524,  5244,  60066,  816474, 13339956, ...];
n=6: [1, 7,  77,  791,  8547, 101619, 1381499, 22023891, ...];
n=7: [1, 8, 100, 1136, 13234, 164528, 2263888, 35555864, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360235, A360236, A360237.
Cf. A360346, A360338.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+2))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+2))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 3.146325060582260657459991059461810..., c = 0.007037477865521004701131626931596125... - Vaclav Kotesovec, Jan 31 2023

A360237 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 7, 87, 1495, 31865, 793769, 22290228, 689397657, 23116772771, 831159921411, 31787496335409, 1285410740283302, 54708408148614317, 2441969507507612684, 113988651908380638224, 5551479742274622439616, 281540748098045175486249, 14843765603832700589293465
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Comments

Sequences with g.f. A(x,k) such that [x^n] A(x,k)^(n+1) = [x^n] (1 + x*A(x,k)^(n+k))^(n+1) have a rate of growth: a(n) ~ c(k) * d^n * n! * n^alfa(k), where d = A360279 = 2.1246065836242897918278825746989... (independent on k) and alfa(k) = 1.256334309718765863868089027485828533429844901971596190707510781... + k*0.94499537543174739679595101598799077876961098786349034... - Vaclav Kotesovec, Feb 05 2023

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 87*x^3 + 1495*x^4 + 31865*x^5 + 793769*x^6 + 22290228*x^7 + 689397657*x^8 + 23116772771*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 8*x^2 + 109*x^3 + 1984*x^4 + 43816*x^5 + 1116182*x^6 + 31810516*x^7 + 994086874*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+5))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 24/3, 436/4, 9920/5, 262896/6, 7813274/7, 254484128/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  7,   87,  1495,  31865,  793769,  22290228, ...];
n=1: [1, 2, 15,  188,  3213,  67938, 1679767,  46874234, ...];
n=2: [1, 3, 24,  304,  5175, 108627, 2666476,  73945899, ...];
n=3: [1, 4, 34,  436,  7403, 154368, 3763020, 103713764, ...];
n=4: [1, 5, 45,  585,  9920, 205626, 4979200, 136401955, ...];
n=5: [1, 6, 57,  752, 12750, 262896, 6325530, 172251150, ...];
n=6: [1, 7, 70,  938, 15918, 326704, 7813274, 211519589, ...];
n=7: [1, 8, 84, 1144, 19450, 397608, 9454484, 254484128, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+5))^(n+1):
n=0: [1, 1,   5,   45,   585,   9920,   205626,   4979200, ...];
n=1: [1, 2,  13,  126,  1654,  27688,   563565,  13415580, ...];
n=2: [1, 3,  24,  253,  3402,  56679,  1135813,  26574702, ...];
n=3: [1, 4,  38,  436,  6065, 101400,  2008616,  46226504, ...];
n=4: [1, 5,  55,  685,  9920, 167686,  3299580,  74828790, ...];
n=5: [1, 6,  75, 1010, 15285, 262896,  5165838, 115758780, ...];
n=6: [1, 7,  98, 1421, 22519, 396109,  7813274, 173599042, ...];
n=7: [1, 8, 124, 1928, 32022, 578320, 11506804, 254484128, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360235, A360236.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+5))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+5))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+5))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.981311186877502847847844107425..., c = 0.000055660090340764345672306890127... - Vaclav Kotesovec, Jan 31 2023

A360235 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+3))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 5, 48, 673, 12057, 256763, 6232909, 168035350, 4945380012, 157008686993, 5331606427775, 192417007138176, 7344652874314128, 295384546093569838, 12478509340848604628, 552330553975194126634, 25560514938260757190962, 1234444956694450007259989, 62114842767595821207341042
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 48*x^3 + 673*x^4 + 12057*x^5 + 256763*x^6 + 6232909*x^7 + 168035350*x^8 + 4945380012*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 6*x^2 + 64*x^3 + 946*x^4 + 17403*x^5 + 375913*x^6 + 9203150*x^7 + 249561291*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+3))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 18/3, 256/4, 4730/5, 104418/6, 2631391/7, 73625200/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  5,  48,  673,  12057,  256763,  6232909, ...];
n=1: [1, 2, 11, 106, 1467,  25940,  546674, 13164522, ...];
n=2: [1, 3, 18, 175, 2397,  41868,  873317, 20861712, ...];
n=3: [1, 4, 26, 256, 3479,  60080, 1240618, 29397424, ...];
n=4: [1, 5, 35, 350, 4730,  80836, 1652870, 38851165, ...];
n=5: [1, 6, 45, 458, 6168, 104418, 2114759, 49309524, ...];
n=6: [1, 7, 56, 581, 7812, 131131, 2631391, 60866723, ...];
n=7: [1, 8, 68, 720, 9682, 161304, 3208320, 73625200, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+3))^(n+1):
n=0: [1, 1,   3,   18,   175,   2397,   41868,   873317, ...];
n=1: [1, 2,   9,   60,   580,   7678,  129842,  2642540, ...];
n=2: [1, 3,  18,  136,  1350,  17520,  287288,  5690016, ...];
n=3: [1, 4,  30,  256,  2661,  34404,  550050, 10593112, ...];
n=4: [1, 5,  45,  430,  4730,  61811,  971600, 18221525, ...];
n=5: [1, 6,  63,  668,  7815, 104418, 1629245, 29869968, ...];
n=6: [1, 7,  84,  980, 12215, 168294, 2631391, 47432554, ...];
n=7: [1, 8, 108, 1376, 18270, 261096, 4125864, 73625200, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360236, A360237.
Cf. A360347, A360338.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+3))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+3))^(n+1) for n >= 0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+3))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 4.09132043601400805425594207544980..., c = 0.00160512950354606176706886534963706... - Vaclav Kotesovec, Jan 31 2023

A360345 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 5, 62, 1214, 31269, 973485, 34993597, 1412846469, 62926155294, 3053566438307, 160005640085764, 8992869671470675, 539298198547460797, 34364052537634696986, 2318526571023659653665, 165143229278977841236029, 12385688813185721332861730, 975844100444710104444582984
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 62*x^3 + 1214*x^4 + 31269*x^5 + 973485*x^6 + 34993597*x^7 + 1412846469*x^8 + 62926155294*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 6*x^2 + 78*x^3 + 1543*x^4 + 39810*x^5 + 1239252*x^6 + 44537587*x^7 + 1798314384*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 18/3, 312/4, 7715/5, 238860/6, 8674764/7, 356300696/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  5,  62,  1214,  31269,   973485,  34993597, ...];
n=1: [1, 2, 11, 134,  2577,  65586,  2025492,  72397390, ...];
n=2: [1, 3, 18, 217,  4104, 103212,  3161648, 112357788, ...];
n=3: [1, 4, 26, 312,  5811, 144428,  4387978, 155030276, ...];
n=4: [1, 5, 35, 420,  7715, 189536,  5710930, 200579975, ...];
n=5: [1, 6, 45, 542,  9834, 238860,  7137401, 249182232, ...];
n=6: [1, 7, 56, 679, 12187, 292747,  8674764, 301023241, ...];
n=7: [1, 8, 68, 832, 14794, 351568, 10330896, 356300696, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n+1))^(n+1):
n=0: [1, 1,   1,    5,    62,   1214,    31269,    973485, ...];
n=1: [1, 2,   7,   42,   479,   8750,   216258,   6562156, ...];
n=2: [1, 3,  18,  136,  1560,  26895,   633608,  18631701, ...];
n=3: [1, 4,  34,  312,  3767,  62888,  1412530,  40031684, ...];
n=4: [1, 5,  55,  595,  7715, 128041,  2763270,  75234930, ...];
n=5: [1, 6,  81, 1010, 14172, 238860,  5016947, 131313798, ...];
n=6: [1, 7, 112, 1582, 24059, 418166,  8674764, 219340759, ...];
n=7: [1, 8, 148, 2336, 38450, 696216, 14466592, 356300696, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360342, A360343, A360344, A360346, A360347.
Cf. A302702, A302703.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n+1))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.635402029299..., c = 0.0308525091280143... - Vaclav Kotesovec, Feb 06 2023

A360236 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 6, 66, 1028, 20138, 464863, 12162876, 351915528, 11075859686, 374858234365, 13530279602015, 517628371405448, 20890826296067329, 886175281852068632, 39393952245422498344, 1830781283537184304756, 88768944166701791039297, 4482797026386165709436753, 235417696462456105986818505
Offset: 0

Views

Author

Paul D. Hanna, Jan 30 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 66*x^3 + 1028*x^4 + 20138*x^5 + 464863*x^6 + 12162876*x^7 + 351915528*x^8 + 11075859686*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 7*x^2 + 85*x^3 + 1401*x^4 + 28339*x^5 + 666638*x^6 + 17651052*x^7 + 514911165*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(n+4))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 21/3, 340/4, 7005/5, 170034/6, 4666466/7, 141208416/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  6,  66,  1028,  20138,  464863,  12162876, ...];
n=1: [1, 2, 13, 144,  2224,  43124,  986694,  25632830, ...];
n=2: [1, 3, 21, 235,  3606,  69264, 1571169,  40527480, ...];
n=3: [1, 4, 30, 340,  5193,  98888, 2224444,  56974172, ...];
n=4: [1, 5, 40, 460,  7005, 132351, 2953185,  75110670, ...];
n=5: [1, 6, 51, 596,  9063, 170034, 3764599,  95085882, ...];
n=6: [1, 7, 63, 749, 11389, 212345, 4666466, 117060623, ...];
n=7: [1, 8, 76, 920, 14006, 259720, 5667172, 141208416, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(n+4))^(n+1):
n=0: [1, 1,   4,   30,   340,   5193,   98888,   2224444, ...];
n=1: [1, 2,  11,   90,  1025,  15330,  284912,   6277922, ...];
n=2: [1, 3,  21,  190,  2220,  32862,  597579,  12884601, ...];
n=3: [1, 4,  34,  340,  4131,  61208, 1094268,  23093756, ...];
n=4: [1, 5,  50,  550,  7005, 104951, 1856360,  38416740, ...];
n=5: [1, 6,  69,  830, 11130, 170034, 2996425,  61005672, ...];
n=6: [1, 7,  91, 1190, 16835, 263956, 4666466,  93880165, ...];
n=7: [1, 8, 116, 1640, 24490, 395968, 7067220, 141208416, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360231, A302702, A302703, A360234, A360235, A360237.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(m+4))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(n+4))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(n+4))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alfa, where d = A360279 = 2.12460658362428979..., alfa = 5.036315811445755451051893091437..., c = 0.000317937301879544729612100255927... - Vaclav Kotesovec, Jan 31 2023

A360344 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 4, 45, 820, 19820, 582007, 19812744, 760177656, 32275309743, 1497313010037, 75208566398988, 4062020902196139, 234638046113989856, 14432573619909530980, 941883830760366274935, 65013065172020161949992, 4733236746727327140204578, 362575149419405494321544263
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 45*x^3 + 820*x^4 + 19820*x^5 + 582007*x^6 + 19812744*x^7 + 760177656*x^8 + 32275309743*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 5*x^2 + 58*x^3 + 1057*x^4 + 25471*x^5 + 746143*x^6 + 25364298*x^7 + 972602305*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 15/3, 232/4, 5285/5, 152826/6, 5223001/7, 202914384/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  4,  45,   820,  19820,  582007,  19812744, ...];
n=1: [1, 2,  9,  98,  1746,  41640, 1212239,  41021862, ...];
n=2: [1, 3, 15, 160,  2790,  65643, 1894300,  63714729, ...];
n=3: [1, 4, 22, 232,  3965,  92028, 2632070,  87984416, ...];
n=4: [1, 5, 30, 315,  5285, 121011, 3429725, 113930075, ...];
n=5: [1, 6, 39, 410,  6765, 152826, 4291758, 141657348, ...];
n=6: [1, 7, 49, 518,  8421, 187726, 5223001, 171278801, ...];
n=7: [1, 8, 60, 640, 10270, 225984, 6228648, 202914384, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n))^(n+1):
n=0: [1, 1,   0,    0,     0,      0,       0,         0, ...];
n=1: [1, 2,   5,   22,   218,   3724,   87245,   2516506, ...];
n=2: [1, 3,  15,   91,   888,  13929,  308182,   8594133, ...];
n=3: [1, 4,  30,  232,  2397,  35712,  742902,  19860536, ...];
n=4: [1, 5,  50,  470,  5285,  77631, 1530000,  38965400, ...];
n=5: [1, 6,  75,  830, 10245, 152826, 2900808,  70300080, ...];
n=6: [1, 7, 105, 1337, 18123, 280140, 5223001, 121085308, ...];
n=7: [1, 8, 140, 2016, 29918, 485240, 9053576, 202914384, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360342, A360343, A360345, A360346, A360347.
Cf. A302702, A302703.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 1.1169011279372..., c = 0.052820142023857... - Vaclav Kotesovec, Feb 06 2023

A360342 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) for n >= 0.

Original entry on oeis.org

1, 1, 2, 20, 316, 6686, 173379, 5255624, 180911070, 6938866748, 292678301988, 13446616806957, 668017569348751, 35678261176871802, 2038906890461704040, 124171127134721710130, 8030684434410398312840, 549848454475826567644385, 39744302449387229743134043
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 20*x^3 + 316*x^4 + 6686*x^5 + 173379*x^6 + 5255624*x^7 + 180911070*x^8 + 6938866748*x^9 + ...
RELATED SERIES.
G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins:
B(x) = 1 + x + 3*x^2 + 27*x^3 + 417*x^4 + 8727*x^5 + 225018*x^6 + 6800714*x^7 + 233778499*x^8 + ... + b(n)*x^n + ...
such that b(n) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) / (n+1),
as well as b(n) = [x^n] A(x)^(n+1) / (n+1),
so that b(n) begin:
[1/1, 2/2, 9/3, 108/4, 2085/5, 52362/6, 1575126/7, 54405712/8, ...].
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n+1) begins:
n=0: [1, 1,  2,  20,  316,  6686,  173379,  5255624, ...];
n=1: [1, 2,  5,  44,  676, 14084,  361794, 10897390, ...];
n=2: [1, 3,  9,  73, 1086, 22266,  566441, 16950588, ...];
n=3: [1, 4, 14, 108, 1553, 31312,  788620, 23442284, ...];
n=4: [1, 5, 20, 150, 2085, 41311, 1029745, 30401460, ...];
n=5: [1, 6, 27, 200, 2691, 52362, 1291355, 37859166, ...];
n=6: [1, 7, 35, 259, 3381, 64575, 1575126, 45848685, ...];
n=7: [1, 8, 44, 328, 4166, 78072, 1882884, 54405712, ...]; ...
Compare to the table of coefficients in (1 + x*A(x)^(2*n-2))^(n+1):
n=0: [1, 1,  -2,   -1,   -32,   -519,  -11490,  -305967, ...];
n=1: [1, 2,   1,    0,     0,      0,       0,        0, ...];
n=2: [1, 3,   9,   28,   180,   2379,   47111,  1182009, ...];
n=3: [1, 4,  22,  108,   745,   8556,  153292,  3658316, ...];
n=4: [1, 5,  40,  265,  2085,  22706,  366450,  8157230, ...];
n=5: [1, 6,  63,  524,  4743,  52362,  781973, 16041192, ...];
n=6: [1, 7,  91,  910,  9415, 109536, 1575126, 29886445, ...];
n=7: [1, 8, 124, 1448, 16950, 211840, 3042820, 54405712, ...]; ...
to see that the main diagonals of the tables are the same.

Links

Crossrefs

Cf. A360343, A360344, A360345, A360346, A360347.
Cf. A360231, A302702.

Programs

  • PARI
    {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(2*m-2))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) for n>=0.
(2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(2*n-2))^(n+1) / (n+1).
a(n) ~ c * d^n * n! * n^alpha, where d = 3.93464558322824528799..., alpha = 0.0798993252137..., c = 0.118957192149397... - Vaclav Kotesovec, Feb 06 2023
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.